# Questions tagged [billiards]

For questions about billiards; dynamical systems involving point particles (billiard balls) that travel in straight lines on the interior of some bounded domain (the billiard table) and experience perfectly elastic reflections on collision with boundary.

65 questions
Filter by
Sorted by
Tagged with
29 views

### Can billiards fill any figure?

For any 2D closed shape, is there always a point and angle for which the billiard movement of a ball fully completes the interior of the shape? In other words, does the path eventually converge to the ...
• 3,427
23 views

### Can ellipsoid or other surface trap incoming light or photon

There had already several questions about light-trapping curves, such as drawing a curve captures light or photon-trapping curve, and well known cases had been answered such as ellipse or hyperbola. ...
• 540
79 views

### Proof that Polygonal Billiards Have Zero Entropy

I'm reading this paper "Billiards In Polygons" by Boldrighini et al. They say that polygonal billiards have zero measure-theoretic entropy, because a given element of the configuration space ...
• 123
49 views

### Billiard problem variation [duplicate]

Suppose a square table. You are shooting from a corner. A ball ends its path upon landing in a corner. At what angles can you shoot a ball from your corner such that the ball will never come back to ...
62 views

### If $D$ isn't a convex billiard table, then the billiard map $T$ is not continuous

I am currently reading chapter 3 of "Geometry and Billiards" by Serge Tabachnikov, and I have some doubts about the need of using convex sets. So, here's how the billiard map is defined: To ...
• 996
8 views

### Construct a mirrored rectangle tunnel unilluminable from one end to another, consisting of only orthogonal rectangles and circles

In a problem there consist of a 2D tunnel, a rectangle with two opposite sides removed as its faces, in which obstacles are only orthogonal rectangles (those whose edges are parallel to one of the ...
137 views

### Reference request: Why is an integrable system called an integrable system and why is the dynamical billiard on a disk completely integrable?

I am seeking detailed reference or references to help me understand the following: Relevant history and motivation behind the term "integrable system" with appropriate primers The meaning ...
• 1,137
25 views

### Elliptical billiards and eccentricity

I have to do an exploration in maths. I was thinking to do it on how the angle of incidence and reflection vary with the eccentricity of the ellipse. Do you think could be interesting? Do you have any ...
1 vote
32 views

### Secure Polygons

The secure-square problem is this: Suppose you have a square with mirror edges. In these square, you choose two points, one as a light source and the other as a target. On these square, you can place ...
• 394
63 views

### Alhazen Billiard problem in 3d

I'm sure that everyone knows what the Alhazen's Billiard problem which is basically determining the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
1 vote
98 views

### Bound on angle implicitly defined via trigonometric equation

Introduction / setup: In the article Spectral Gap for a Class of Random Billiards by Renato Feres and Hong-Kun Zhang (link: https://www.math.wustl.edu/~feres/spectralgapnew.pdf) a trigonometric ...
• 73
1 vote
184 views

### Trigonometric relation between angles in billiard dynamics

Introduction: In the article Spectral Gap for a Class of Random Billiards by Renato Feres and Hong-Kun Zhang (link: https://www.math.wustl.edu/~feres/spectralgapnew.pdf ), proposition 8 (page 20) does ...
• 73
74 views

### Why a measure over the points of a billiard which reflect infinitely often in finite time vanish, i.e. why $\int_{N\cap M}f(x^{-})d\mu_1(x^{-1})=0$

The crux of my question is that why $\mu(N^{(2)}) = 0$ when $N^{(2)}$ consists all points of a billiard that reflect infinitely often in finite time under a given flow (i.e. transformation) function. ...
• 976
105 views

### For arbitrary billiard tables with elastic boundary reflections, is the "Lebesgue measure" an invariant of the flow maps?

This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) ...
1 vote
59 views

### Particular values for the sum of divisors function from billiards

In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article. I've wondered if we can compute some simple ...
1 vote
42 views

### Motivation for the definition of a compact Riemann manifold with piecewise smooth boundary

I am asking for either motivation on the requirement regarding $f_i^{-1}(0)$ in the following definition, or better yet a reference to a book dealing with this subject. The following is the definition ...
• 976
224 views

### Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit

In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is: How can I reproduce Schwartz's result ...
• 351
873 views

### Proving a billiards bank shot procedure gives equal angles of incidence and reflection

One method of making a bank shot in billiards involves imagining two lines, the so called "cross-pocket" and "cross-ball" lines: One then projects their point of intersection to ...
1 vote
137 views

### Law of reflection when tangent is undefined

The law of reflection also holds for non-plane mirrors, provided that the normal at any point on the mirror is understood to be the outward pointing normal to the local tangent plane of the mirror at ...
• 61
158 views

### Application of billiards

Studying billiards is a difficult problem in general, even in pretty simple cases it has plenty of interesting properties. I would like to understand what can be applications (mathematically or in ...
• 1,077
224 views

### Rectangular billiards: ergodic or not?

I'm approaching dynamical billiards theory, and I've found that rectangular/square billiards are among the examples of integrable and non-ergodic ones. But in this contribution it is (resolutely) said ...
• 769
109 views

### Sinai billiard isomorphic to hard spheres problem

I have heard that Sinai's famous results on the ergodicity of $n$ hard spheres defined on a $d$ dimensional hypercube with periodic boundaries is isomorphic to a billiard (from which it was proved), ...
• 200
184 views

### Phase space and collision map of a billiard

I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions. 2.5. Phase space for the flow. The state of a ...
• 200
227 views

### Number of rays intersecting at a point inside a polygon

I'm working on a project to do with bouncing rays inside polygons and now I've reached a crucial stage of this project in which I need help with and is related to the problem stated below. Your help ...
• 93
1 vote
114 views

• 3,133
407 views

### Texts on Mathematical Billiards

I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains ...
• 115
45 views

### Verifying that a billiard system satisfies the twist condition.

I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre. ...
99 views

### Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
170 views

### Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
• 30.3k
1 vote
81 views

### Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
36 views

### Joining boundary points through affine/billiards trajectories

Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path ...
• 1,185
179 views

### Billiard ball mental patients.

The Question: Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital. Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$ metres ...
• 45.1k
1 vote
186 views

### Unfolding a billiard trajectory on an unbounded billiard table

A common "trick" used in the study of polygonal billiards is to unfold the trajectory. I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, ...
• 65
1 vote
67 views

### Reflecting a vector within a box (variation on the billiards problem)

Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers. A starting point is chosen at random within the box at (x,y), such ...
• 19
1k views

### The simplest billiards problem

The Problem: Let's say we have a rectangle of size $m \times n$ centered at the origin (or, if it makes the math easier, you can place it wherever on the plane). We take a billiard ball, ...
• 691
125 views

### Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
• 309
1 vote
175 views

### Trajectories in circular Billards

Given two points $p_1,p_2$ on a circular billard table. I want to know all billard trajectories from $p_1$ to $p_2$ hitting the boundary precisely once. Model of the circular billard: Denote the ...
• 516
228 views

• 649